Exponential and Logarithmic Models

Explore compound growth, half-life, logarithmic time.

Lesson 5 of 53 parts Practice Flashcards

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Compound Growth

Objective: Explain compound growth and use it to solve or interpret a problem.

A quantity compounded at fixed intervals follows a repeated multiplier, with a rate and time unit stated. Compare model predictions to context and data. The factor applied during each time interval.

Worked examples

Try itExponential and Logarithmic Models — practice 1.1: Start at 4 and multiply by 2 each period. Find the value after 5 periods.Show answer

Step 1: Use yₜ=4·2^t. Step 2: At t=5, compute 2^5=32. Step 3: Multiply by 4 to get 128.

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Compound Growth

Objective: Explain compound growth and use it to solve or interpret a problem.

A quantity compounded at fixed intervals follows a repeated multiplier, with a rate and time unit stated. Compare model predictions to context and data. The factor applied during each time interval.

Worked examples

Try itExponential and Logarithmic Models — practice 1.1: Start at 4 and multiply by 2 each period. Find the value after 5 periods.Show answer

Step 1: Use yₜ=4·2^t. Step 2: At t=5, compute 2^5=32. Step 3: Multiply by 4 to get 128.

Half-Life

Objective: Explain half-life and use it to solve or interpret a problem.

After n half-life periods, a remaining population is A₀(1/2)^n in the ideal model. It describes a population trend, not a scheduled time for each item. One quarter of the initial amount.

Worked examples

Try itExponential and Logarithmic Models — practice 2.1: Start at 1280 and halve for 8 periods. What remains?Show answer

Step 1: Use the factor (1/2)^8. Step 2: Divide 1280 by 2^8=256. Step 3: The remaining amount is 5.

Logarithmic Time

Objective: Explain logarithmic time and use it to solve or interpret a problem.

If A·b^t reaches target T with A>0 and b>0,b≠1, solve t=log_b(T/A) when T/A>0; interpret units and limits. The real logarithm argument must be positive.

Worked examples

Try itExponential and Logarithmic Models — practice 3.1: Evaluate log_4(4194304).Show answer

Step 1: By definition, log_4(y)=x iff 4^x=y. Step 2: Here 4^11=4194304. Step 3: The logarithm equals 11.

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